23.1.164 problem 164

Internal problem ID [4771]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 164
Date solved : Tuesday, September 30, 2025 at 08:31:01 AM
CAS classification : [_linear]

\begin{align*} x y^{\prime }&=a x -\left (-b \,x^{2}+1\right ) y \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 25

ode:=x*diff(y(x),x) = a*x-(-b*x^2+1)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{\frac {b \,x^{2}}{2}} c_1 b -a}{x b} \]

✓ Mathematica. Time used: 0.049 (sec). Leaf size: 30

ode=x*D[y[x],x]==a*x-(1-b*x^2)*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {-a+b c_1 e^{\frac {b x^2}{2}}}{b x} \end{align*}

✓ Sympy. Time used: 0.245 (sec). Leaf size: 53

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-a*x + x*Derivative(y(x), x) + (-b*x**2 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \begin {cases} \frac {C_{1} e^{\frac {b x^{2}}{2}}}{x} - \frac {a}{b x} & \text {for}\: b \neq 0 \\\text {NaN} & \text {otherwise} \end {cases}, \ y{\left (x \right )} = \begin {cases} \frac {C_{1}}{2 b x^{3} - 6 x} - \frac {3 a x^{2}}{2 b x^{3} - 6 x} & \text {for}\: b = 0 \\\text {NaN} & \text {otherwise} \end {cases}\right ] \]